L(s) = 1 | + 2·3-s − 2·4-s + 5-s + 5·7-s + 9-s − 11-s − 4·12-s + 4·13-s + 2·15-s + 4·16-s − 17-s − 2·20-s + 10·21-s + 6·23-s + 25-s − 4·27-s − 10·28-s + 3·29-s − 2·31-s − 2·33-s + 5·35-s − 2·36-s − 8·37-s + 8·39-s − 9·41-s − 4·43-s + 2·44-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 4-s + 0.447·5-s + 1.88·7-s + 1/3·9-s − 0.301·11-s − 1.15·12-s + 1.10·13-s + 0.516·15-s + 16-s − 0.242·17-s − 0.447·20-s + 2.18·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.88·28-s + 0.557·29-s − 0.359·31-s − 0.348·33-s + 0.845·35-s − 1/3·36-s − 1.31·37-s + 1.28·39-s − 1.40·41-s − 0.609·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 337535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 337535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.14314022405773, −12.40525390556128, −11.92909824406248, −11.34515675944559, −10.99789305459936, −10.42881082212656, −10.13519947318214, −9.385058582939643, −8.994322355059762, −8.557307648990058, −8.373835011055167, −8.110259159423708, −7.358833348447747, −7.013615437455234, −6.224501994795310, −5.552727197679769, −5.046160429903008, −4.997073023310432, −4.159617170787514, −3.766803932689087, −3.231079783224661, −2.623773535534817, −1.977544089545450, −1.426947637383424, −1.084688290102732, 0,
1.084688290102732, 1.426947637383424, 1.977544089545450, 2.623773535534817, 3.231079783224661, 3.766803932689087, 4.159617170787514, 4.997073023310432, 5.046160429903008, 5.552727197679769, 6.224501994795310, 7.013615437455234, 7.358833348447747, 8.110259159423708, 8.373835011055167, 8.557307648990058, 8.994322355059762, 9.385058582939643, 10.13519947318214, 10.42881082212656, 10.99789305459936, 11.34515675944559, 11.92909824406248, 12.40525390556128, 13.14314022405773